A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies

Nasser Sweilam; Fathalla Rihan; Seham AL-Mekhlafi

doi:10.3934/dcdss.2020120

Article Contents

2020, Volume 13, Issue 9: 2403-2424. Doi: 10.3934/dcdss.2020120

This issue Previous Article Bistability analysis of virus infection models with time delays Next Article Dynamics of solutions of a reaction-diffusion equation with delayed inhibition

A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies

1.
Department of Mathematics, Faculty of Science, Cairo University, Egypt
2.
Department of Mathematical Sciences, United Arab Emirates University, Al-Ain, 15551, UAE
3.
Department of Mathematics, Faculty of Education, Sana'a University, Yemen

^* Corresponding author: Nasser Sweilam
^* Corresponding author: Nasser Sweilam

Received: October 2018

Revised: December 2019

Early access: October 31, 2019

Published online: October 31, 2019

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Abstract

In this paper, we present an optimal control problem of fractional-order delay-differential model for cancer treatment based on the synergy between anti-angiogenic and immune cells therapies. The governed model consists of eighteen differential equations. A discrete time-delay is incorporated to represent the time required for the immune system to interact with the cancer cells, and fractional-order derivative is considered to reflect the memory and hereditary properties in the process. Two control variables for immunotherapy and anti-angiogenic therapy are considered to reduce the load of cancer cells. Necessary conditions that guarantee the existence and the uniqueness of the solution for the control problem have been considered. We approximate numerically the solution of the optimal control problem by solving the state system forward and adjoint system backward in time. Some numerical simulations are provided to validate the theoretical results.

Keywords:

Mathematics Subject Classification: 92C50, 93A30, 37N25, 37N35, 34K28.

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Figure 1. Numerical simulations of the state variables with two controls treatment cases and different values of $ d_{\tau} $, $ \alpha = 0.96 $ using IOCM

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Figure 2. Numerical simulations of the control variables and different values of $ d_{\tau} $, $ \alpha = 0.96 $ using IOCM

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Figure 3. Relation between the state variables when $ d_{\tau} = 3 $ and $ \alpha = 0.98 $ using IOCM with two controls treatment cases

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Figure 4. Numerical simulations of the control variables and different values of $ \alpha $ and $ d_{\tau} = 1 $, using IOCM

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Figure 5. The relationship between the variables $ T(t-d_{\tau}), I(t-d_{\tau}), V(t-d_{\tau}) $ and $ R(t-d_{\tau}) $ and $ T(t), I(t), V(t) $ and $ R(t) $ values of $ \alpha = 0.92 $ and $ d_{\tau} = 2 $ using IOCM

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Figure 6. The relationship between the variables T(t − d_τ), I(t − d_τ), V (t − d_τ) and R(t − d_τ) and T(t), I(t), V (t) and R(t) values of α = 0.92 and d_τ = 2 using IOCM

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Table 1. The parameters of system (2)-(19) and their descriptions

Parameters of fraction power	Descriptions
$ \gamma^{\alpha} $	The tumor growth rate
$ a^{\alpha} $	The antigenicity for tumor
$ \lambda^{\alpha} $, $ \delta_{u}^{\alpha} $, $ \delta_{D}^{\alpha} $, $ I_{1}^{\alpha} $, $ R_{1}^{\alpha} $	The parameters for dendritic cell expansion
$ \alpha_{A_{1}}^{\alpha} $	Max growth rate for Ang-1
$ \delta_{A_{1}}^{\alpha} $	Degradation rate for Ang-1
$ \alpha_{A_{2}}^{\alpha} $	Max growth rate for Ang-2
$ \delta_{A_{2}}^{\alpha} $	Degradation rate for Ang-2
$ \theta_{A_{2}} $	$ \frac{1}{2} $ max cancer cells needed to trigger $ A_{2} $ production
$ \alpha_{v} $	Rate $ VEGF $ is constantly expressed by glioma cells
$ \alpha_{v_{2}} $	Max growth rate of $ VEGF $ production
$ \delta_{v}^{\alpha} $	Degradation rate of $ VEGF $
$ \alpha_{y}^{\alpha} $	Proliferation rate of endothelial cells
$ \delta_{y}^{\alpha} $	Apototic rate of endothelial cells
$ s^{\alpha} $	Conversion factor from microvessels to $ ECs $
$ \gamma_{B}^{\alpha} $	Max rate microvessels break down to $ ECs $
$ \rho^{\alpha} $	$ \frac{1}{2} $ max $ \frac{VEGF}{EC} $ needed to cause regression, growth, etc.
$ \omega^{\alpha} $	Max rate $ ECs $ mature to microvessels
$ \theta_{y}^{\alpha} $	$ \frac{1}{2} $ max $ \frac{VEGF}{EC} $ needed to keep $ ECs $ alive
$ \theta_{v_{a}}^{\alpha} $	$ \frac{1}{2} $ max $ \frac{VEGF}{EC} $ needed to induce $ EC $ cell cycle
$ \theta_{EC}^{\alpha} $	$ \frac{1}{2} $ max $ \frac{A2}{A1} $ ratio where $ A2 $ blocks tie-2 receptor from $ A1 $
$ \theta_{B}^{\alpha} $	$ \frac{1}{2} $ max $ \frac{A1}{A2} $ ratio where $ A1 $ matures vessels
$ \tau^{\alpha} $	Binding rate of anti-$ VEGF $ antibody with $ VEGF $
$ \rho_{v_{a}}^{\alpha} $	Degradation rate of anti-$ VEGF $ antibody
$ r_{0}^{\alpha} $, $ k_{2}^{\alpha} $, $ k_{3}^{\alpha} $, $ s_{1}^{\alpha} $	Parameters for tumor progression
$ \alpha_{i}^{\alpha}, $ $ i=1,...,7 $ $ M^{\alpha} $
$ k_{4}^{\alpha}, $ $ C_{1}^{\alpha} $, $ s_{j}^{\alpha} $, $ j=1,2 $, $ \delta_{A}^{\alpha} $,	Parameters for T-Cell expansions
$ \delta_{E}^{\alpha} $, $ \delta_{H}^{\alpha} $, $ \delta_{R}^{\alpha} $
$ p_{c}^{\alpha} $, $ p_{1}^{\alpha} $, $ p_{2}^{\alpha} $, $ p_{4}^{\alpha} $,	Parameters of IL-2, TGF-$ \beta $ and IL-10 concentrations
$ I_{2}^{\alpha} $, $ s_{4}^{\alpha} $, $ \tau_{c}^{\alpha} $, $ \tau_{s}^{\alpha} $, $ \tau^{\alpha} $

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Table 2. The Parameter values of system (2)-(19) [38]

$ Parameter $	The value of Parameter
$ a^{\alpha} $	$ (10^{-5} day^{-1})^{\alpha} $
$ C^{\alpha} $	$ (0.3 ng^{-1}mL)^{\alpha} $
$ I_{1}^{\alpha} $	$ (0.4 ng^{-1}mL)^{\alpha} $
$ I_{2}^{\alpha} $	$ (0.75 ng^{-1}mL)^{\alpha} $
$ K_{2}^{\alpha} $	$ (1.2 )^{\alpha} $
$ K_{3}^{\alpha} $	$ (11)^{\alpha} $
$ K_{4}^{\alpha} $	$ (0.33 )^{\alpha} $
$ M^{\alpha} $	$ (10^{7} cell)^{\alpha} $
$ M_{E}^{\alpha} $	$ (3\times10^{6} cell)^{\alpha} $
$ M_{H}^{\alpha} $	$ (6\times10^{6} cell)^{\alpha} $
$ M_{R}^\alpha $	$ (1\times10^{6} cell)^{\alpha} $
$ p_{1}^{\alpha} $	$ (1.8\times10^{-8} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{2}^{\alpha} $	$ (1.1\times10^{-7} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{3}^{\alpha} $	$ (1.4\times10^{-8} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{4}^{\alpha} $	$ (1.3\times10^{-10} ng^{-1}mL day^{-1} cell^{-1})^{\alpha} $
$ p_{c}^{\alpha} $	$ (1.5\times10^{-7} ng^{-1}mLday^{-1} cell^{-1})^{\alpha} $
$ r_{0}^{\alpha} $	$ (0.9 day^{-1})^{\alpha} $
$ R_{1}^{\alpha} $	$ (2\times 10^{7})^{\alpha} $
$ s^{\alpha} $	$ (0.7 \frac{EC}{\mu M})^{\alpha} $
$ S_{1}^{\alpha} $	$ (3.5 ng mL^{-1})^{\alpha} $
$ S_{2}^{\alpha} $	$ (2.9 ng mL^{-1})^{\alpha} $
$ S_{3}^{\alpha} $	$ (1.7 ng mL^{-1})^{\alpha} $
$ S_{4}^{\alpha} $	$ (0.9 ng mL^{-1})^{\alpha} $
$ V_{1}^{\alpha} $	$ (3.5 ng mL^{-1})^{\alpha} $
$ V_{2}^{\alpha} $	$ (2.9 ng mL^{-1})^{\alpha} $
$ V_{3}^{\alpha} $	$ (0.14 ng mL^{-1})^{\alpha} $
$ \alpha_{1}^{\alpha} $	$ (23 day^{-1})^{\alpha} $
$ \alpha_{2}^{\alpha} $	$ (16 day^{-1})^{\alpha} $
$ \alpha_{3}^{\alpha} $	$ (9.9 day^{-1})^{\alpha} $
$ \alpha_{4}^{\alpha} $	$ (1.9 day^{-1})^{\alpha} $
$ \alpha_{5}^{\alpha} $	$ (5.1 day^{-1})^{\alpha} $
$ \alpha_{6}^{\alpha} $	$ (2.1 day^{-1})^{\alpha} $
$ \alpha_{7}^{\alpha} $	$ (0.022 day^{-1})^{\alpha} $
$ \alpha_{A_{1}}^{\alpha} $	$ (0.24 \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{A_{2}}^{\alpha} $	$ (1.92 \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{V}^{\alpha} $	$ (3\times 10^{-6} \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{V_{2}}^{\alpha} $	$ (3.7\times10^{-2} \frac{ng}{M\times\mu\times day})^{\alpha} $
$ \alpha_{Y}^{\alpha} $	$ (0.198 day^{-1})^{\alpha} $
$ \gamma_{1}^{\alpha} $	$ (2.1 day^{-1})^{\alpha} $
$ \gamma_{B}^{\alpha} $	$ (0.8 day^{-1})^{\alpha} $
$ \theta_{A_{2}}^{\alpha} $	$ (10^{6} cell)^{\alpha} $
$ \theta_{B}^{\alpha} $	$ (1)^{\alpha} $

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Table 3. Comparisons between the value of objective functional using IOCM with and without controls cases and $ T_{f} = 100 $, $ d_{\tau} = 2 $

$ \alpha $	$ J(u_A^{\ast}, u_M^{\ast}) $ without control	$ J(u_A^{\ast}, u_M^{\ast}) $ with Two controls
1	$ 7.7864 \times 10^{8} $	$ 9.402886\times 10^{4} $
0.90	$ 1.90928 \times 10^{9} $	$ 8.9381\times 10^{4} $
0.80	$ 1.7003\times 10^{9} $	$ 8.1454\times 10^{4} $
0.70	$ 9.0261 \times 10^{8} $	$ 6.7976\times 10^{4} $
0.60	$ 2.1615\times 10^{8} $	$ 4.5080\times 10^{4} $
0.50	$ 1.8270\times 10^{7} $	$ 2.9438\times 10^{4} $
0.40	$ 1.5197\times 10^{6} $	$ 1.9119\times 10^{3} $
0.30	$ 2.5346\times 10^{5} $	$ 855.8715 $

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Table 4. The value of objective functional $ J(u_A^{\ast}, u_M^{\ast}) $, $ T_{f} = 50 $ with different value of $ d_{\tau} $ and control case

$ d_{\tau} $	$ J(u_A^{\ast}, u_M^{\ast}) $
0	$ 4.2401\times 10^4 $
1	$ 4.2419\times 10^4 $
3	$ 4.2447\times 10^4 $
5	$ 4.2467\times 10^4 $
10	$ 4.2504\times 10^4 $

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Table 5. Comparisons between IOCM, GEM and $ d_{\tau} = 1, $ $ T_{f} = 10. $

$ \alpha $	Methods	$ J(u_A^{\ast}, u_M^{\ast}) $
1	IOCM	$ 3.2798 \times 10^3 $
	GEM	$ 8.8514 \times 10^3 $
0.98	IOCM	$ 2.6575 \times 10^3 $
	GEM	$ 1.2839 \times 10^4 $
0.90	IOCM	$ 1.1506 \times 10^3 $
	GEM	$ 7.1531 \times 10^4 $
0.80	IOCM	$ 414.5745 $
	GEM	$ 1.4226\times 10^6 $

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